Linear estimates for solutions of quadratic equations in free groups

TL;DR

This paper establishes bounds on the lengths of solutions to quadratic equations in free groups, providing explicit bounds for orientable and non-orientable cases based on coefficient lengths.

Contribution

It introduces explicit length bounds for solutions of quadratic equations in free groups, distinguishing between orientable and non-orientable cases.

Findings

Length of variable values in solutions is bounded by 2s for orientable equations.

Length is bounded by 12s^4 for non-orientable equations.

Provides a quantitative measure for solutions in free groups.

Abstract

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by $2s$ for orientable equation and by $12s^4$ for non-orientable equation, where $s$ is the sum of the lengths of the coefficients